Is the topic of ratio and proportion important? Socrates thought so. In fact, is is so important that we use the term ratio to describe thought when it is sane and sound: rational thought. Proportion is, in fact, the basis of all algebraic equations. Here is Socrates' in conversation with several lads where he explains why we all need to learn ratio and proportion

The text is from Jowett's translation of Plato's Republic. The numbers in brackets are reference numbers you will find in the margin or text of any translation of the Republic (they are called "Stephanus numbers").

An illustration will make my meaning clearer: -- here are three fingers -- a little finger, a second finger, and a middle finger.

Very good.

You may suppose that they are seen quite close: And here comes the point.

What is it?

Each of them equally [523d] appears a finger, whether seen in the middle or at the extremity, whether white or black, or thick or thin -- it makes no difference; a finger is a finger all the same. In these cases a man is not compelled to ask of thought the question what is a finger? for the sight never intimates to the mind that a finger is other than a finger.

True.

And therefore, I said, as we might expect, there is nothing here which invites or excites [523e] intelligence.

There is not, he said.

But is this equally true of the greatness and smallness of the fingers? Can sight adequately perceive them? and is no difference made by the circumstance that one of the fingers is in the middle and another at the extremity? And in like manner does the touch adequately perceive the qualities of thickness or thinness, of softness or hardness? And so of the other senses; do they give perfect intimations of such matters? Is not their mode of operation on this wise -- [524a] the sense which is concerned with the quality of hardness is necessarily concerned also with the quality of softness, and only intimates to the soul that the same thing is felt to be both hard and soft?

You are quite right, he said.

And must not the soul be perplexed at this intimation which the sense gives of a hard which is also soft? What, again, is the meaning of light and heavy, if that which is light is also heavy, and that which is heavy, light?

[524b]
Yes, he said, these intimations which the soul receives are very curious and require to be explained.

Yes, I said, and in these perplexities the soul naturally summons to her aid calculation and intelligence, that she may see whether the several objects announced to her are one or two.

True.

And if they turn out to be two, is not each of them one and different?

Certainly.

And if each is one, and both are two, she will conceive the two as in a state of division, for if they were undivided [524c] they could only be conceived of as one?

True.

The eye certainly did see both small and great, but only in a confused manner; they were not distinguished.

Yes.

Whereas the thinking mind, intending to light up the chaos, was compelled to reverse the process, and look at small and great as separate and not confused.

Very true.

Was not this the beginning of the enquiry "What is great?" and "What is small?"

Exactly so.

And thus arose the distinction of the visible and the intelligible.

[524d]
Most true.

This was what I meant when I spoke of impressions which invited the intellect, or the reverse -- those which are simultaneous with opposite impressions, invite thought; those which are not simultaneous do not.

I understand, he said, and agree with you.

And to which class do unity and number belong?

I do not know, he replied.

Think a little and you will see that what has preceded will supply the answer; for if simple unity could be adequately perceived by the sight [524e] or by any other sense, then, as we were saying in the case of the finger, there would be nothing to attract towards being; but when there is some contradiction always present, and one is the reverse of one and involves the conception of plurality, then thought begins to be aroused within us, and the soul perplexed and wanting to arrive at a decision asks [525a] "What is absolute unity?" This is the way in which the study of the one has a power of drawing and converting the mind to the contemplation of true being.

And surely, he said, this occurs notably in the case of one; for we see the same thing to be both one and infinite in multitude?

Yes, I said; and this being true of one must be equally true of all number?

Certainly.

And all arithmetic and calculation have to do with number?

[525b]
Yes.

And they appear to lead the mind towards truth?

Yes, in a very remarkable manner.